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Theorem sbalex 2242
Description: Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2065.

That both sides of the biconditional express proper substitution is proved by sb5 2276 and sb6 2085. The implication "to the left" is equs4v 1999 and does not require ax-10 2141 nor ax-12 2177. It also holds without disjoint variable condition if we allow more axioms (see equs4 2421). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2465 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2464 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2268 in place of equsex 2423 in order to remove dependency on ax-13 2377. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2065. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.)

Assertion
Ref Expression
sbalex (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sbalex
StepHypRef Expression
1 nfe1 2150 . . 3 𝑥𝑥(𝑥 = 𝑡𝜑)
2 ax12ev2 2180 . . 3 (∃𝑥(𝑥 = 𝑡𝜑) → (𝑥 = 𝑡𝜑))
31, 2alrimi 2213 . 2 (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
4 equs4v 1999 . 2 (∀𝑥(𝑥 = 𝑡𝜑) → ∃𝑥(𝑥 = 𝑡𝜑))
53, 4impbii 209 1 (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784
This theorem is referenced by:  equsexvOLD  2269  sb5  2276  dfsb7  2279  alexeqg  3651  dfdif3OLD  4118  pm13.196a  44433
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